PART I: MATHEMATICAL PREREQUISITES. REVIEW OF TOPICS FROM ANALYSIS. Sets and Functions. Intervals and Inequalities. Mathematical Induction. Polynomials and Partial Fractions. Remarks on the Statement of Theorems. Complex Numbers in Cartesian Form. Complex Numbers in Polar Form. Roots. Some Properties of Integrals. Linear Difference Equations. PART II: VECTORS AND LINEAR ALGEBRA. ALGEBRA OF VECTORS. Preliminaries. Scalars and Vectors. Vectors-A Geometrical Approach in R3. Vectors in Component Form. Scalar Product (Dot Product). Vector Product (Cross Product). Combinations of Scalar and Vector Products. Geometrical Applications of Scalar and Vector Products. Vector Spaces. MATRICS. Introductory Ideas. Addition of Matrices, Multiplication by a Number and the Transposition Operation. Matrix Multiplication. Linear Transformations. Differentiation. Systems of Linear Equations: Solution by Elimination. Linear Independence. Rank. Reduced Echelon Form. Systems of Linear Equations: Existence and Form of Solution in Terms of Rank. Determinants. Determinants and Rank. Cramer's Rule. Inverse Matrices. Algebraic Eigenvalue Problems. Eigenvalues. Diagonalizability of Matrices. The Cayley-Hamilton Theorem. Quadratic Forms. The LU and Cholesky Factorization Methods. PART III: ORDINARY DIFFERENTIAL EQUATIONS. FIRST ORDER ORDINARY DIFFERENTIAL EQUATIONS. Differential Equations and Their Origins. First Order Differential Equations and Isoclines. Separable Equations. Exact Differential Equations and Integrating Factors. Linear First Order Differential Equations. Orthogonal and Isogonal Trajectories. Existence, Uniqueness and an Iterative Method of Solution. Numerical Solution of First Order Equations by the Runge-Kutta Method. LINEAR HIGHER ORDER ORDINARY DIFFERENTIAL EQUATIONS. Linear Higher Order Ordinary Differential Equations. Second Order Constant Coefficient Equations-Homogeneous Case. Higher Order Constant Coefficient Equations-Homogeneous Case. Differential Operators. Nonhomogeneous Linear Differential Equations. General Reduction of the Order of a Linear Differential Equation. Integral Method. Oscillatory Behaviour. Reduction to the Normal Form u"-f(x)u=0. The Green's Function. SYSTEMS OF LINEAR DIFFERENTIAL EQUATIONS. First Order Linear Homogeneous System of Differential Equations. First Order Linear Nonhomogeneous Systems of Differential Equations. Second Order Linear Systems of Differential Equations. Qualitative Theory: The Phase Plane and Stability. Numerical Solution of Systems by the Runge-Kutta Method. LAPLACE TRANSFORM AND z-TRANSFORM. The Laplace Transform-Introductory Ideas. Operational Properties of the Laplace Transform. Applications of the Laplace Transform. The z-Transform. Applications of the z-Transform. SERIES SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS. Sequences, Convergence and Power Series. Solving Differential Equations by Taylor Series. Solution in the Neighbourhood of an Ordinary Point. Legendre's Equation and Legendre Polynomials. The Gamma Function t(x). Frobenius' Method and Its Extension. Bessel Functions. Asymptotic Expansions. Numerical Solution of Second Order Equations by the Runge-Kutta Method. FOURIER SERIES, STURM-LIOUVILLE PROBLEMS AND ORTHOGONAL FUNCTIONS. Trigonometric Series, Periodic Extension and Convergence. The Formal Development of Fourier Series. Convergence of Fourier Series and Related Results. Integration and Differentiation of Fourier Series. Conditions for the Uniform Convergence of a Fourier Series. Estimating the Fourier Coefficients-Smoothness and Its Effect on Convergence. Numerical Harmonic Analysis. Representation of Functions Using Orthogonal Systems. Sturm-Liouville Problems. Expansions in Terms of Bessel Functions. Orthogonal Polynomials. APPENDICES. The Laplace Transform. The z-Transform. The Gamma Function t(x). Bessel Functions J0(x), J1(x), Y0(x) and Y1(x). ANSWERS TO ODD-NUMBERED PROBLEMS. SUGGESTED READING AND REFERENCES LIST. INDEX.
